Seminars and Colloquia by Series

Monday, September 26, 2011 - 16:05 , Location: Skiles 006 , Robert Krone , Georgia Tech , Organizer: Anton Leykin
An ideal of a local polynomial ring can be described by calculating astandard basis with respect to a local monomial ordering. However if we areonly allowed approximate numerical computations, this process is notnumerically stable. On the other hand we can describe the ideal numericallyby finding the space of dual functionals that annihilate it. There areseveral known algorithms for finding the truncated dual up to any specifieddegree, which is useful for zero-dimensional ideals. I present a stoppingcriterion for positive-dimensional cases based on homogenization thatguarantees all generators of the initial monomial ideal are found. This hasapplications for calculating Hilbert functions.
Monday, September 19, 2011 - 16:05 , Location: Skiles 006 , Saikat Biswas , Georgia Tech , Organizer: Josephine Yu
The rational solutions to the equation describing an elliptic curve form a finitely generated abelian group, also known as the Mordell-Weil group. Detemining the rank of this group is one of the great unsolved problems in mathematics. The Shafarevich-Tate group of an elliptic curve is an important invariant whose conjectural finiteness can often be used to determine the generators of the Mordell-Weil group. In this talk, we first introduce the definition of the Shafarevich-Tate group. We then discuss the theory of visibility, initiated by Mazur, by means of which non-trivial elements of the Shafarevich-Tate group of an elliptic curve an be 'visualized' as rational points on an ambient curve. Finally, we explain how this theory can be used to give theoretical evidence for the celebrated Birch and Swinnerton-Dyer Conjecture.
Wednesday, May 25, 2011 - 15:05 , Location: Skiles 006 , Don Zagier , MPI Bonn and College de France , Organizer: Stavros Garoufalidis
Come and see!
Wednesday, April 13, 2011 - 13:00 , Location: Skiles 006 , Christelle Vincent , University of Wisconsin Madison , Organizer: Matt Baker
For q a power of a prime, consider the ring \mathbb{F}_q[T]. Due to the many similarities between \mathbb{F}_q[T] and the ring of integers \mathbb{Z}, we can define for \mathbb{F}_q[T] objects that are analogous to elliptic curves, modular forms, and modular curves. In particular, for \mathfrak{p} a prime ideal in \mathbb{F}_q[T], we can define the Drinfeld modular curve X_0(\mathfrak{p}), and study the reduction modulo \mathfrak{p} of its Weierstrass points, as is done in the classical case by Rohrlich, and Ahlgren and Ono. In this talk we will present some partial results in this direction, defining all necessary objects as we go. The first 20 minutes should be accessible to graduate students interested in number theory.
Monday, April 11, 2011 - 15:00 , Location: Skiles 005 , Fernando Rodriguez-Villegas , University of Texas Austin , Organizer: Matt Baker
We will discuss several instances of sequences of complex manifolds X_n whose Betti numbers b_i(X_n) converge, when properly scaled, to a limiting distribution. The varieties considered have Betti numbers which are described in a combinatorial way making their study possible. Interesting examples include varieties X for which b_i(X) is the i-th coefficient of the reliability polynomial of an associated graph.
Thursday, March 31, 2011 - 16:00 , Location: Skiles 006 , Pete Clark , University of Georgia , Organizer: Matt Baker
Which commutative groups can occur as the ideal class group (or "Picard group") of some Dedekind domain?  A number theorist naturally thinks of the case of integer rings of number fields, in which the class group must be finite and the question of which finite groups occur is one of the deepest in algebraic number theory.  An algebraic geometer naturally thinks of affine algebraic curves, and in particular, that the Picard group of the standard affine ring of an elliptic curve E over C is isomorphic to the group of rational points E(C), an uncountably infinite (Lie) group.  An arithmetic geometer will be more interested in Mordell-Weil groups, i.e., E(k) when k is a number field -- again, this is one of the most notorious problems in the field.  But she will at least be open to the consideration of E(k) as k varies over all fields. In 1966, L.E. Claborn (a commutative algebraist) solved the "Inverse Picard Problem": up to isomorphism, every commutative group is the Picard group of some Dedekind domain.  In the 1970's, Michael Rosen (an arithmetic geometer) used elliptic curves to show that any countable commutative group can serve as the class group of a Dedekind domain.  In 2008 I learned about Rosen's work and showed the following theorem: for every commutative group G there is a field k, an elliptic curve E/k and a Dedekind domain R which is an overring of the standard affine ring k[E] of E -- i.e., a domain in between k[E] and its fraction field k(E) -- with ideal class group isomorphic to G.  But being an arithmetic geometer, I cannot help but ask about what happens if one is not allowed to pass to an overring: which commutative groups are of the form E(k) for some field k and some elliptic curve E/k?  ("Inverse Mordell-Weil Problem") In this talk I will give my solution to the "Inverse Picard Problem" using elliptic curves and give a conjectural answer to the "Inverse Mordell-Weil Problem".  Even more than that, I can (and will, time permitting) sketch a proof of my conjecture, but the proof will necessarily gloss over a plausible technicality about Mordell-Weil groups of "arithmetically generic" elliptic curves -- i.e., I do not in fact know how to do it.  But the technicality will, I think, be of interest to some of the audience members, and of course I am (not so) secretly hoping that someone there will be able to help me overcome it.
Thursday, March 31, 2011 - 15:00 , Location: Skiles 006 , Xander Faber , University of Georgia , Organizer: Matt Baker
Given a nonconstant holomorphic map f: X \to Y between compact Riemann surfaces, one of the first objects we learn to construct is its ramification divisor R_f, which describes the locus at which f fails to be locally injective. The divisor R_f is a finite formal linear combination of points of X that is combinatorially constrained by the Hurwitz formula. Now let k be an algebraically closed field that is complete with respect to a nontrivial non-Archimedean absolute value. For example, k = C_p. Here the role of a Riemann surface is played by a projective Berkovich analytic curve. As these curves have many points that are not algebraic over k, some new (non-algebraic) ramification behavior appears for maps between them. For example, the ramification locus is no longer a divisor, but rather a closed analytic subspace. The goal of this talk is to introduce the Berkovich projective line and describe some of the topology and geometry of the ramification locus for self-maps f: P^1 \to P^1.
Thursday, March 17, 2011 - 16:00 , Location: Skiles 006 , Joe Rabinoff , Harvard University , Organizer: Matt Baker
An elliptic curve over the integer ring of a p-adic field whose special fiber is ordinary has a canonical line contained in its p-torsion.  This fact has many arithmetic applications: for instance, it shows that there is a canonical partially-defined section of the natural map of modular curves X_0(Np) -> X_0(N).  Lubin was the first to notice that elliptic curves with "not too supersingular" reduction also contain a canonical order-p subgroup.  I'll begin the talk by giving an overview of Lubin and Katz's theory of the canonical subgroup of an elliptic curve.  I'll then explain one approach to defining the canonical subgroup of any abelian variety (even any p-divisible group), and state a very general existence result.  If there is time I'll indicate the role tropical geometry plays in its proof.
Thursday, March 17, 2011 - 15:00 , Location: Skiles 006 , Kirsten Wickelgren , Harvard University , Organizer: Matt Baker
Grothendieck's anabelian conjectures say that hyperbolic curves over certain fields should be K(pi,1)'s in algebraic geometry. It follows that points on such a curve are conjecturally the sections of etale pi_1 of the structure map. These conjectures are analogous to equivalences between fixed points and homotopy fixed points of Galois actions on related topological spaces. This talk will start with an introduction to Grothendieck's anabelian conjectures, and then present a 2-nilpotent real section conjecture: for a smooth curve X over R with negative Euler characteristic, pi_0(X(R)) is determined by the maximal 2-nilpotent quotient of the fundamental group with its Galois action, as the kernel of an obstruction of Jordan Ellenberg. This implies that the set of real points equipped with a real tangent direction of the smooth compactification of X is determined by the maximal 2-nilpotent quotient of Gal(C(X)) with its Gal(R) action, showing a 2-nilpotent birational real section conjecture.
Monday, March 14, 2011 - 15:00 , Location: Skiles 005 , Patrick Ingram , University of Waterloo , Organizer: Matt Baker
In classical holomorphic dynamics, rational self-maps of the Riemann sphere whose critical points all have finite forward orbit under iteration are known as post-critically finite (PCF) maps. A deep result of Thurston shows that if one excludes examples arising from endomorphisms of elliptic curves, then PCF maps are in some sense sparse, living in a countable union of zero-dimensional subvarieties of the appropriate moduli space (a result offering dubious comfort to number theorists, who tend to work over countable fields). We show that if one restricts attention to polynomials, then the set of PCF points in moduli space is actually a set of algebraic points of bounded height. This allows us to give an elementary proof of the appropriate part of Thurston's result, but it also provides an effective means of listing all PCF polynomials of a given degree, with coefficients of bounded algebraic degree (up to the appropriate sense of equivalence).